Dimension of a Set of Transformations

[jsgoodfella]

If we consider the set R of all linear transformations from an p-dimensional vector space Z to Z (T:Z -> Z), what do we know about the dimension of the set R?

In other words, what do we know about any basis for R? What are its properties?

 

[micromass]

Hint: those linear mappings can be represented by matrices. What's the dimension of this space of matrices?

 

[jsgoodfella]

Hint: those linear mappings can be represented by matrices. What's the dimension of this space of matrices?

Well, we know that each matrix is pxp, because Z is p-dimensional. The dimension of this space of matrices is p-squared (for instance there are 4 basis matrices for the 2x2 case).
But how can we know that each matrix in the p^2 dimensional basis is a linear matrix?

 

[micromass]

But how can we know that each matrix in the p^2 dimensional basis is a linear matrix?

Huh, what do you mean?? Every matrix is linear.

 

[jsgoodfella]

Thanks for staying with me!

Huh, what do you mean?? Every matrix is linear.

I'm thinking of the general basis for p^2 - If p is two, then [{0,1};{0,0}] could be a matrix in the basis which doesn't represent a linear transformation. Do we just assume they are all linear?

On another note, is there a linear transformation that maps every vector in a vector space to zero?

 

[micromass]

I'm thinking of the general basis for p^2 - If p is two, then [{0,1};{0,0}] could be a matrix in the basis which doesn't represent a linear transformation. Do we just assume they are all linear?

It does represent a linear function, namely f(x,y)=(y,0). Every matrix represents a linear function, that's why matrices are so important.

On another note, is there a linear transformation that maps every vector in a vector space to zero?

Yep, that is a linear transformation. It's the analog of the zero matrix.

 

[jsgoodfella]

Thanks for your help!

Just to make sure, the dimension is p-squared, right?

 

[micromass]

Right!